Theoretical and computational issues for improving … · Theoretical and computational issues for...

Theoretical and computational issues forimproving the performance of linear

optimization methods

Pedro Munari

Advisor: Marcos Nereu Arenales (ICMC/USP)

Co-advisor: Jacek Gondzio (University of Edinburgh)

Theoretical and computational issues for improving the performance of linear optimization methodsPedro Munari [[email protected]] - XXXV Congresso Nacional de Matematica Aplicada e Computacional (CNMAC 2014)

Applied Mathematics: What we do

I We are interested in solving problems which are relevant in our

day-to-day lives;

I Model: mathematical formulation for a real-life problem;

I We can apply all the theoretical and computational tools to analyse

and solve the model;

I Formal, reliable and safe way to support decision-making.

I We cannot stop until the problem is completely under our domain!

day-to-day lives;

Applied Mathematics: What we actually do

day-to-day lives;

I We are interested in solving complex and difficult problems which

are relevant in our day-to-day lives!

I Propose and improve models for real-life problems!

I We want to apply, improve and propose theoretical and

computational tools to analyse and solve the models!

I Formal, reliable, safe, effective and efficient way to decision-making!

Optimization

I One of the tools in Applied Mathematics;

I Get the best solution from a set of possible ones;

I Natural idea, we are trying to optimize all the time;

I Mathematical formulation:

minimize f(x) subject to x ∈ X .(maximize)

Optimization

minimize f(x) subject to x ∈ X .

(maximize)

Optimization

Linear Optimization

I f(x) is a linear function;

I X is described by a set of linear equalities/inequalities;

I X may be continuous or discrete;

I Linear Programming: continuous linear formulations;

I Integer Programming: discrete linear formulations;

I Combinatorial Optimization problems.

Linear Optimization

I Although these formulations seem much simpler than the others,

they are able to model very complex situations, from our day-to-day

lives and faced by many companies around the world;

I In addition, the solution methods work relatively well in practice.

Linear Optimization

Production planning

. Lot sizing problem

1 2 3 4 5 6 7 8 9 10 11 12

Demand of product 1 for the next twelve months

1 2 3 4 5 6 7 8 9 10 11 12

Production planning

1 2 3 4 5 6 7 8 9 10 11 12

Production planning

1 2 3 4 5 6 7 8 9 10 11 12

Production planning

1 2 3 4 5 6 7 8 9 10 11 12

Production planning

minn∑

T∑t=1

citxit +

n∑i=1

T∑t=1

hitIit +

n∑i=1

T∑t=1

s.t. xit + Ii,t−1 = dit + Iit, i = 1, . . . , n; t = 1, . . . , T,n∑

(aixit + stiyit) ≤ bt, t = 1, . . . , T,

xit ≤ Cyit, i = 1, . . . , n; t = 1, . . . , T,

Ii0 = 0, i = 1, . . . , n,

xit ≥ 0, Iit ≥ 0, yit ∈ {0, 1}, i = 1, . . . , n; t = 1, . . . , T.

Logistics

. Vehicle routing problem

Logistics

d 1 d 2

c02c03

Logistics

I Find a set of routes to visit all the customers in order to meet the

demand, while minimizing the total travel cost.

I Decision variables :

xijk =

1, vehicle k visit i and goes to j immediately,

0, otherwise.i, j = 0, . . . , n+ 1; k = 1, . . . ,K.

wik : time instant that vehicle k starts servicing the customer i.

i = 0, . . . , n+ 1; k = 1, . . . ,K.

Logistics

xijk =

0, otherwise.i, j = 0, . . . , n+ 1; k = 1, . . . ,K.

i = 0, . . . , n+ 1; k = 1, . . . ,K.

Logistics

xijk =

0, otherwise.i, j = 0, . . . , n+ 1; k = 1, . . . ,K.

i = 0, . . . , n+ 1; k = 1, . . . ,K.

Logistics

xijk =

0, otherwise.i, j = 0, . . . , n+ 1; k = 1, . . . ,K.

i = 0, . . . , n+ 1; k = 1, . . . ,K.

minK∑

n+1∑i=0

n+1∑j=0

cijxijk

s.t.K∑

n+1∑j=1j 6=i

xijk = 1, i = 1, . . . , n,

n∑i=0i6=h

xihk =

n+1∑j=1j 6=h

xhjk, h = 1, . . . , n, k = 1, . . . , K,

n∑i=1

n+1∑j=1j 6=i

xijk ≤ Qk, k = 1, . . . , K,

wjk ≥ wik + (si + tij)xijk + Mij(xijk − 1), i = 0, . . . , n, j = 1, . . . , n + 1, k = 1, . . . , K,

n+1∑j=1

x0jk = 1, k = 1, . . . , K,

n∑i=0

xi,n+1,k = 1, k = 1, . . . , K,

ai ≤ wik ≤ bi, i = 1, . . . , n + 1, k = 1, . . . , K,

xijk ∈ {0, 1}, i, j = 0, . . . , n + 1, k = 1, . . . , K.

Logistics

I Although this formulation is correct, the current methods may take

hours, days or even more, to solve real-life instances;

I By decomposing the problem we can obtain a more efficient

strategy, but still solving the problem may require a long time;

I A way to overcome this is to recur to approximate solutions

(heuristics);

I Or, we can try to improve the current methods and propose new

solution strategies!

Logistics

(heuristics);

Logistics

(heuristics);

Logistics

(heuristics);

Thesis

. Purpose

I Study the state-of-the-art methodologies used to solve linear

optimization problems and see if we can find a way to improve them;

I Simplex method, column generation, branch-price-and-cut;

I Why not using the advantages offered by interior point methods?

Thesis

. Purpose

Thesis

. Purpose

minK∑

n+1∑i=0

n+1∑j=0

cijxijk

s.aK∑

n+1∑j=1j 6=i

xijk = 1, i = 1, . . . , n,

n∑i=0i6=h

xihk =

n+1∑j=1j 6=h

xhjk, h = 1, . . . , n, k = 1, . . . , K,

n∑i=1

n+1∑j=1j 6=i

xijk ≤ Qk, k = 1, . . . , K,

wjk ≥ wik + (si + tij)xijk + Mij(xijk − 1), i = 0, . . . , n, j = 1, . . . , n + 1, k = 1, . . . , K,

n+1∑j=1

x0jk = 1, k = 1, . . . , K,

n∑i=0

xi,n+1,k = 1, k = 1, . . . , K,

ai ≤ wik ≤ bi, i = 1, . . . , n + 1, k = 1, . . . , K,

xijk ∈ {0, 1}, i, j = 0, . . . , n + 1, k = 1, . . . , K.

Special structure of the model

ℤn ℝnMaster

Subproblems

ℤn ℝnMaster

Subproblems

Dantzig-Wolfe Decomposition

I Compact formulation:

min cTx,

s.t. Ax = b, (linking constraints)

Dx = d, (special structure)

x ∈ Zn+,

I x ∈ Zn+: vector of decision variables;

I c ∈ Rn, A ∈ Rm×n, b ∈ Rm, D ∈ Rh×n, d ∈ Rh.

min cTx,

s.t. Ax = b, (linking constraints)

Dx = d, (special structure)

x ∈ Zn+,

min cTx,

s.t. Ax = b,

x ∈ X ,

I X = {x ∈ Zn+ : Dx = d} is a discrete set;

Resolution Theorem. Let X = {x ∈ R | Ax ≥ b} be a nonempty

polyhedron with at least one extreme point. Let [pq]q∈Q be the extreme

points, and let [pr]r∈R be a complete set of extreme rays of X, where Q

and R are the respective sets of indices. Let

∑q∈Q

λqpq +∑r∈R

µrpr|∑q∈Q

λq = 1, λq ≥ 0, µr ≥ 0

Then C = X.

Also known as Representation Theorem and Caratheodory’s Theorem

(see e.g. Bertsimas and Tsitsiklis, 1997).

I Convex hull of X : C = conv(X );

I From the Resolution Theorem, any x ∈ C can be rewritten as a

convex combination of the extreme points and rays of C:

x =∑q∈Q

λqpq +∑r∈R

µrpr,∑q∈Q

λq = 1, λq ≥ 0, µr ≥ 0.

We can use this to replace x in the original problem:

min cTx,

s.t. Ax = b,

x ∈ X .

x =∑q∈Q

λqpq +∑r∈R

µrpr,∑q∈Q

λq = 1, λq ≥ 0, µr ≥ 0.

min cTx,

s.t. Ax = b,

x ∈ X .

x =∑q∈Q

λqpq +∑r∈R

µrpr,∑q∈Q

λq = 1, λq ≥ 0, µr ≥ 0.

min cTx,

s.t. Ax = b,

x ∈ X .

I We obtain the equivalent problem:

min∑q∈Q

λq(cT pq) +∑r∈R

µr(cT pr),

s.t.∑q∈Q

λq(Apq) +∑r∈R

µr(Apr) = b,

∑q∈Q

λq = 1,

λq ≥ 0, µr ≥ 0, ∀q ∈ Q,∀r ∈ R,

x =∑q∈Q

λqpq +∑r∈R

µrpr,

x ∈ Zn+.

where cj = cT pj and aj = Apj , ∀j ∈ Q and ∀j ∈ R.

min∑q∈Q

cqλq +∑r∈R

s.t.∑q∈Q

aqλq +∑r∈R

arµr = b,

∑q∈Q

λq = 1,

λq ≥ 0, µr ≥ 0, ∀q ∈ Q,∀r ∈ R,

x =∑q∈Q

λqpq +∑r∈R

µrpr,

x ∈ Zn+,

with cj = cT pj and aj = Apj , ∀j ∈ Q and ∀j ∈ R.

min∑q∈Q

cqλq +∑r∈R

s.t.∑q∈Q

aqλq +∑r∈R

arµr = b,

∑q∈Q

λq = 1,

λq ≥ 0, µr ≥ 0, ∀q ∈ Q,∀r ∈ R,

x =∑q∈Q

λqpq +∑r∈R

µrpr,

x ∈ Zn+,

A huge number of variables! (one for each extreme point and ray)

I Linear relaxation of the problem:

min∑q∈Q

cqλq +∑r∈R

s.t.∑q∈Q

aqλq +∑r∈R

arµr = b,

∑q∈Q

λq = 1,

λq ≥ 0, µr ≥ 0, ∀q ∈ Q,∀r ∈ R.

I Although we have a huge number of columns, we know how to

generate them!

I Linear relaxation of the problem:

min∑q∈Q

cqλq +∑r∈R

s.t.∑q∈Q

aqλq +∑r∈R

arµr = b,

∑q∈Q

λq = 1,

λq ≥ 0, µr ≥ 0, ∀q ∈ Q,∀r ∈ R.

I Although we have a huge number of columns, we know how to

generate them!

Column generation technique

First paper: Ford and Fulkerson (1958)

I Multicommodity network flow problem;

I Number of variables too large to be dealt explicitly;

I Idea: Change the pricing operation in the simplex method;

I “Treat non-basic variables implicitly”.

Column generation technique

First paper: Ford and Fulkerson (1958)

I Multicommodity network flow problem;

I Number of variables too large to be dealt explicitly;

I Idea: Change the pricing operation in the simplex method;

I “Treat non-basic variables implicitly”.

Column generation

I We are interested in solving a linear programming problem, called

the Master Problem (MP):

z? := min∑j∈N

cjλj ,

s.t.∑j∈N

ajλj = b,

λj ≥ 0, ∀j ∈ N.

I N is too big;

I The columns (cj , aj) are not known explicitly;

I We know how to generate them!

Column generation

z? := min∑j∈N

cjλj ,

s.t.∑j∈N

ajλj = b,

λj ≥ 0, ∀j ∈ N.

I N is too big;

Column generation

z? := min∑j∈N

cjλj ,

s.t.∑j∈N

ajλj = b,

λj ≥ 0, ∀j ∈ N.

I N is too big;

Column generation

I The Restricted Master Problem (RMP):

zRMP := min∑j∈N

cjλj ,

s.t.∑j∈N

ajλj = b,

λj ≥ 0, ∀j ∈ N.

I with N ⊂ N .

I Let (λ, u) be a primal-dual optimal solution of the RMP.

Column generation

zRMP := min∑j∈N

cjλj ,

s.t.∑j∈N

ajλj = b,

λj ≥ 0, ∀j ∈ N.

I with N ⊂ N .

Column generation

zRMP := min∑j∈N

cjλj ,

s.t.∑j∈N

ajλj = b, (u)

λj ≥ 0, ∀j ∈ N.

I with N ⊂ N .

Column generation

I Solution λ of the RMP ⇒ solution λ of the MP;

I λj = λj , if j ∈ N ;

I λj = 0, otherwise;

I cT λ ≥ cTλ?;

I How to know if λ is optimal in the MP?

I Call the pricing subproblem (oracle):

zSP := min{0, cj − uT aj |(cj , aj) ∈ A}.

I (cj , aj) are the variables in the subproblem;

I If zSP < 0, then new columns are generated;

I Otherwise, an optimal solution of the MP was found!

Column generation

I cT λ ≥ cTλ?;

Column generation

I cT λ ≥ cTλ?;

Column generation

I cT λ ≥ cTλ?;

Column generation

I cT λ ≥ cTλ?;

Column generation

I cT λ ≥ cTλ?;

Column generation

I cT λ ≥ cTλ?;

Column generation

INITIAL SOLUTION

ORACLE

STOPPING CRITERION

FIRST COLUMN(S)

UPPER BOUND &

DUAL VARIABLE(S)

NEW COLUMN(S)

LOWER BOUND &

NEW COLUMN(S)

OPTIMAL SOLUTION MASTER PROBLEM

Column generation

I To simplify notation, we assume that X is bounded;

z? := min∑j∈N

cjλj ,

s.t.∑j∈N

ajλj = b,

λj ≥ 0, ∀j ∈ N.

I N is too big;

Column generation

I To get integer solutions, we typically need to combine column

generation with the branch-and-bound method;

I This results in the branch-and-price method;

I In addition, if we add valid inequalities to the MP, then we have a

branch-price-and-cut method;

Column generation

Branch-and-price and branch-price-and-cut

In summary

To solve difficult optimization problems, we typically need:

I Dantzig-Wolfe decomposition: obtain a formulation with a huge number

of variables;

I Column generation: solve the linear relaxation treating variables

(columns) implicitly;

I Start with a subset (RMP) and generate more (Subproblem);

I Branch-and-price: column generation within branch-and-bound in order to

find an integer solution for the original problem;

I Branch-price-and-cut: column and cut generation within

branch-and-bound.

Right! But where are the gaps?

In summary

of variables;

branch-and-bound.

In summary

of variables;

branch-and-bound.

In summary

of variables;

branch-and-bound.

In summary

of variables;

branch-and-bound.

In summary

of variables;

branch-and-bound.

In summary

of variables;

branch-and-bound.

Standard column generation

I Optimal solutions obtained by the simplex method:

⇒ Extreme points of the RMP;

I They oscillate too much between consecutive iterations;

⇒ uj+1 is typically far from uj ;

I Instability and slow convergence of the method.

RMP (dual)

RMP(dual)

Oscillation in a real instance

‖uj − uj+1‖2, for each iteration j:��

��

� � � ��

��

Column generation variants

I Stabilization techniques: avoid extreme solutions!

⇒ use a point in the interior of the feasible set;

I Most of them: modify the master problem!

I Add variables, bounds, constraints, penalties, ...

⇒ The master problem may become more difficult to solve;

⇒ Some of them may be difficult to implement;

⇒ Several parameters to set.

I Stability center: prohibit the next dual solution to go far from it;

RMP (dual)

I The column generation is more efficient when based on well-centered

interior points;

I So, why not using an interior point method?

I This is straightforward: does not require any changes in the RMP

nor parameter adjustments;

I Interior point methods will provided naturally stable solutions.

interior points;

Interior point method

Optimality conditions

KKT conditions:

b−Ax = 0

c−ATu− s = 0

xisi = 0, ∀i = 1, . . . , n

x ≥ 0

s ≥ 0

I µ→ 0 iteratively;

I Instead of strictly satisfying these conditions, the iterates belong to

a safe neighbourhood, e.g.:

N2(θ) = {(x, u, s) ∈ F0 | ‖XSe− µe‖2 ≤ θµ}; or

Ns(γ) = {(x, u, s) ∈ F0 | γµ ≤ xisi ≤ 1γµ, ∀i = 1, . . . , n}, γ ∈ (0, 1).

Primal-dual interior point method

Perturbed KKT conditions:

b−Ax = 0

c−ATu− s = 0

xisi = µ, ∀i = 1, . . . , n

x ≥ 0

s ≥ 0

b−Ax = 0

c−ATu− s = 0

xisi = µ, ∀i = 1, . . . , n

x ≥ 0

s ≥ 0

b−Ax = 0

c−ATu− s = 0

xisi = µ, ∀i = 1, . . . , n

x ≥ 0

s ≥ 0

N2(θ) = {(x, u, s) ∈ F0 | ‖XSe− µe‖2 ≤ θµ};

b−Ax = 0

c−ATu− s = 0

xisi = µ, ∀i = 1, . . . , n

x ≥ 0

s ≥ 0

Simplex method

Non-optimal solutions from interior point method

Advantages:

I We save time, as we stop early;

I The solution is well-centered in the feasible set;

I Early termination: good sub-optimal solutions.

I The column corresponds to a deeper cut in the dual space.

Advantages:

Primal-dual column generation method (PDCGM)

I Primal-dual interior point method to get primal-dual solutions;

I Suboptimal solution (λ, u) (ε-optimal solution): we stop the interior point

method with optimality tolerance ε.

I The distance to optimality ε is dynamically adjusted according to the

relative gap;

ε = min{εmax, gap/D}

I gap = (UB− LB)/(1 + |UB|);

I D: degree of optimality (fixed);

relative gap;

I gap = (UB− LB)/(1 + |UB|);

relative gap;

I gap = (UB− LB)/(1 + |UB|);

relative gap;

I gap = (UB− LB)/(1 + |UB|);

relative gap;

I gap = (UB− LB)/(1 + |UB|);

relative gap;

I gap = (UB− LB)/(1 + |UB|);

Well-centred solution

I (λ, u) should be well-centered in the feasible set:

γµ ≤ (cj − uTaj)λj ≤ (1/γ)µ, ∀j ∈ N,

for some γ ∈ (0.1, 1], where µ = (1/|N |)(cT − uTA)λ;

I Natural way of stabilizing dual solutions if a primal-dual interior

point method is used.

Well-centred solution

I (λ, u) should be well-centered in the feasible set:

γµ ≤ (cj − uTaj)λj ≤ (1/γ)µ, ∀j ∈ N,

for some γ ∈ (0.1, 1], where µ = (1/|N |)(cT − uTA)λ;

I Natural way of stabilizing dual solutions if a primal-dual interior

point method is used.

Answer from the oracle

I The oracle (subproblem) is called with u:

zSP (u) := min{cj − uTaj |(cj , aj) ∈ A}.

I Two cases:

I zSP (u) < 0 and new columns are generated;

I zSP (u) = 0 and no columns are generated;

I The lower bound provided by a suboptimal solution is still valid?

I If zSP (u) = 0, does the method terminate?

I Two cases:

Convergence

Let zSP = zSP (u) be the value of the oracle corresponding to the

suboptimal solution (λ, u). Then, κzSP + bT u ≤ z?.

Let (λ, u) be the suboptimal solution of the RMP, found at iteration k

with tolerance εk > 0. If zSP = 0, then the new relative gap is strictly

smaller than the previous one, i.e., gapk < gapk−1.

Convergence

Let zSP = zSP (u) be the value of the oracle corresponding to the

suboptimal solution (λ, u). Then, κzSP + bT u ≤ z?.

Let (λ, u) be the suboptimal solution of the RMP, found at iteration k

with tolerance εk > 0. If zSP = 0, then the new relative gap is strictly

smaller than the previous one, i.e., gapk < gapk−1.

PDCGM: Algorithm

1. Input: Initial RMP; parameters κ, εmax, D > 1, δ > 0, .

2. set LB = −∞, UB =∞, gap =∞, ε = 0.5;

3. while (gap > δ) do

4. find a well-centered ε-optimal solution (λ, u) of the RMP;

5. UB = min{UB, zRMP };

6. call the oracle with the query point u;

7. LB = max{LB, κzSP + bT u};

8. gap = (UB− LB)/(1 + |UB|);

9. ε = min{εmax, gap/D};

10. if (zSP < 0) then add the new columns into the RMP;

11. end(while)33

PDCGM: Algorithm

2. set LB = −∞, UB =∞, gap =∞, ε = 0.5;

8. gap = (UB− LB)/(1 + |UB|);

11. end(while)33

PDCGM: Algorithm

2. set LB = −∞, UB =∞, gap =∞, ε = 0.5;

8. gap = (UB− LB)/(1 + |UB|);

11. end(while)33

PDCGM: Proof of convergence

Theorem

Let z? be the optimal value of the MP. Given the optimality tolerance

δ > 0, the primal-dual column generation method converges in a finite

number of steps to a primal feasible solution λ of the MP with

objective value z that satisfies:

(z − z?) < δ(1 + |z|).

Idea of the proof: By using the previous Lemmas, we show that either new

columns are generated or the relative gap is strictly reduced at each iteration,

until it falls below the optimality tolerance.

PDCGM: Proof of convergence

Theorem

Let z? be the optimal value of the MP. Given the optimality tolerance

δ > 0, the primal-dual column generation method converges in a finite

number of steps to a primal feasible solution λ of the MP with

objective value z that satisfies:

(z − z?) < δ(1 + |z|).

Idea of the proof: By using the previous Lemmas, we show that either new

columns are generated or the relative gap is strictly reduced at each iteration,

until it falls below the optimality tolerance.

PDCGM: source code

I Implementation in C, using interior point solver HOPDM;

I Publicly available code

http://www.maths.ed.ac.uk/~gondzio/software/pdcgm.html

I Source-code examples are provided for 6 different applications:

I Cutting stock problem;

I Vehicle routing problem;

I Capacitated lot sizing problem;

I Multiple kernel learning;

I Two-stage stochastic programming;

I Multicommodity network flow.

PDCGM: source code

Number of iterations

Sheet2

CPU time

Outer iterations

182.33

120.19

290.02

34.5225.93

126.01

SCGMPDCGMACCPM

CLSPST

SCGMPDCGMACCPM

CLSPST

Relative to PDCGM CSP VRPTW CLSPST

SCGM 1.52 1.33 1.60

ACCPM 2.41 4.86 1.26

CPU time (s)

Sheet2

CPU time

Outer iterations

182.33

120.19

290.02

34.5225.93

126.01

SCGMPDCGMACCPM

CLSPST

SCGMPDCGMACCPM

CLSPST

Relative to PDCGM CSP VRPTW CLSPST

SCGM 3.50 1.95 1.26

ACCPM 8.97 4.01 1.27

Oscillation in a VRPTW instance (Solomon C207)

‖uj − uj+1‖2, for each iteration j:��

��

� � � ��

��

PDCGM: Remarks

I Using well-centered, suboptimal solutions is beneficial for column

generation;

I The computational study was based on linear relaxations;

I Next step: branch-and-price method!

PDCGM: Remarks

generation;

PDCGM: Remarks

generation;

PDCGM: Remarks

generation;

Branch-and-price and Branch-price-and-cut

Interior point branch-price-and-cut (IPBPC)

I Several challenging issues are involved when using this algorithm

within a branch-price-and-cut framework;

I It is not just replacing a simplex-type method.

I Change of strategy!

I Rethink every piece of a standard BPC: column generation, valid

inequalities, branching, ...

I The primal-dual interior point algorithm will be used to provide

well-centered, suboptimal solutions:

I Column generation;

I Valid inequalities;

I Branching.

I More stable primal and dual solutions;

I Deeper columns and cuts;

I Speed up the solution times.

I Branching.

Computational experiments

I Vehicle Routing Problem with Time Windows (VRPTW)

I The IPBPC performance was compared to the best results that are

available in the literature for a simplex-based BPC:

I Desaulniers, Lessard and Hadjar, Transp. Science, 2008.

Computational experiments

I Vehicle Routing Problem with Time Windows (VRPTW)

I The IPBPC performance was compared to the best results that are

available in the literature for a simplex-based BPC:

I Desaulniers, Lessard and Hadjar, Transp. Science, 2008.

Theoretical and computational issues for improving the performance of linear optimization methodsPedro Munari [[email protected]] - XXXV Congresso Nacional de Matematica Aplicada e Computacional (CNMAC 2014)Nodes_100

0 20 40 60 80 100 120

Number of nodes

DLH08 IPBPC

Comparing to a simplex-based BPC

Number of nodes

DLH08 IPBPC Ratio

C1 9 9 1.00

RC1 104 78 1.33

R1 239 182 1.31

352 269 1.31

Theoretical and computational issues for improving the performance of linear optimization methodsPedro Munari [[email protected]] - XXXV Congresso Nacional de Matematica Aplicada e Computacional (CNMAC 2014)Cuts_100

0 100 200 300 400 500 600 700 800

Number of valid inequalities

DLH08 IPBPC

Valid inequalities

DLH08 IPBPC Ratio

C1 0 0 1.00

RC1 2199 1191 1.85

R1 3391 2140 1.58

5590 3331 1.68

CPUtime_100

0 2000 4000 6000 8000 10000 12000 14000 16000 18000

CPU time

DLH08 IPBPC

Seconds

CPU time (sec)

DLH08 IPBPC Ratio

C1 158 28 5.69

RC1 17198 3472 4.95

R1 27928 4621 6.04

45284 8121 5.58

200-series instances:

I Wider time windows and larger vehicle capacity;

I More difficult subproblems!

Theoretical and computational issues for improving the performance of linear optimization methodsPedro Munari [[email protected]] - XXXV Congresso Nacional de Matematica Aplicada e Computacional (CNMAC 2014)Nodes_200

0 2 4 6 8 10 12 14 16 18

Number of nodes

DLH08 IPBPC

Number of nodes

DLH08 IPBPC Ratio

C2 8 8 1.00

RC2 12 10 1.20

R2 38 42 0.90

58 60 0.97

Cuts_200

0 50 100 150 200 250 300 350 400

DLH08 IPBPC

Valid inequalities

DLH08 IPBPC Ratio

C2 0 0 1.00

RC2 456 166 2.75

R2 1336 525 2.54

1792 691 2.59

Theoretical and computational issues for improving the performance of linear optimization methodsPedro Munari [[email protected]] - XXXV Congresso Nacional de Matematica Aplicada e Computacional (CNMAC 2014)CPUtime_200

0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000

400904

CPU time

DLH08 IPBPC

Seconds

CPU time (sec)

DLH08 IPBPC Ratio

C2 16745 865 19.35

RC2 92365 3250 28.42

R2 504540 27166 18.57

613650 31281 19.62

Discrete Optimization

New developments in the primal–dual column generation technique

Jacek Gondzio a, Pablo González-Brevis a,⇑,1, Pedro Munari b,2

a School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, United Kingdomb Instituto de Ciências Matemáticas e de Computação, University of São Paulo, Av. Trabalhador São-carlense, 400, Centro, Cx. Postal 668, CEP 13560-970 São Carlos, SP, Brazil

a r t i c l e i n f o

Article history:

Received 6 October 2011

Accepted 18 July 2012

Available online 31 July 2012

a b s t r a c t

The optimal solutions of the restricted master problems typically leads to an unstable behavior of the

standard column generation technique and, consequently, originates an unnecessarily large number of

iterations of the method. To overcome this drawback, variations of the standard approach use interior

points of the dual feasible set instead of optimal solutions. In this paper, we focus on a variation known

European Journal of Operational Research 224 (2013) 41–51

Contents lists available at SciVerse ScienceDirect

European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Using the primal-dual interior point algorithm within

the branch-price-and-cut method

Pedro Munari a,n,1, Jacek Gondzio b

a Instituto de Ciencias Matematicas e de Computac- ~ao, University of S ~ao Paulo, Av. Trabalhador, S ~ao-carlense, 400 - Centro, Cx. Postal 668, CEP 13560-970,

S ~ao Carlos-SP, Brazilb School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ,

United Kingdom

a r t i c l e i n f o

Available online 14 March 2013

Keywords:

a b s t r a c t

Branch-price-and-cut has proven to be a powerful method for solving integer programming problems.

It combines decomposition techniques with the generation of both columns and valid inequalities and

Contents lists available at SciVerse ScienceDirect

journal homepage: www.elsevier.com/locate/caor

Computers & Operations Research

Computers & Operations Research 40 (2013) 2026–2036

Thesis

I Linear optimization is a very rich research area;

I Strong theoretical development and powerful methodologies;

I Still, many gaps to be closed...

I I focused on new ideas for the simplex method, column generation

and branch-and-price, applied to real-life problems.

I Using the advantages offered by interior point methods was essential

for column generation and branch-and-price;

I Combining these methods was a challenge!

Thesis

Currently

I Professor at UFSCar;

I Production Engineering Department;

I Projects on interior point branch-price-and-cut and variants of

vehicle routing problems;

I VRP with multiple deliverymen;

I VRP under uncertainty; (in collaboration with Univ. of Edinburgh)

I Aircraft assignment - air taxi;

I Crop rotation scheduling with sustainable restrictions.

I I need soldiers :)

Currently

Acknowledgements

I Advisor: Prof. Dr. Marcos Arenales;

I Co-advisor: Prof. Dr. Jacek Gondzio;

I Earlier advisor: Prof. Dr. Geraldo Nunes Silva;

I Colleagues and professors at ICMC/USP;

I SBMAC;

I CNMAC organizers;

I Examination committee of “Premio Odelar Leite Linhares 2014”.

Acknowledgements

I SBMAC;

I CNMAC organizers;

Acknowledgements

I SBMAC;

I CNMAC organizers;

Acknowledgements

I SBMAC;

I CNMAC organizers;

Acknowledgements

I SBMAC;

I CNMAC organizers;

Acknowledgements

I SBMAC;

I CNMAC organizers;

Acknowledgements

I SBMAC;

I CNMAC organizers;

I Thank you!

I Questions?

I Funding

I [email protected]

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